Question

Find nth fibonacci number binets formula

Answer 1

We have only defined the nth Fibonacci number in terms of the two before it:the n-th Fibonacci number is the sum of the (n-1)th and the (n-2)th.So to calculate the 100th Fibonacci number, for instance, we need to compute all the 99 values before it first - quite a task, even with a calculator! 
A natural question to ask therefore is:Can we find a formula for F(n) which involves 
only n and does not need any other (earlier) Fibonacci values?Yes! It involves our golden section number Phi and its reciprocal phi:
Here it is:
Fib(n) =  Phin − (−Phi)−n  =   Phin − ( −phi)n√5√5
On these pages we usePhi = √5 + 1  =   1·61803 39887 49894 84820 ... and phi = Phi −1 = 1 =√5 −1 = 0·61803 39887 49894 84820 ... 2Phi2The next version uses just one of the golden section values: Phi, and all the powers are positive:Fib(n) = Phin –(–1)n
Phin
5

Since phi is the name we use for 1/Phi on these pages, then we can remove the fraction in the numerator here and make it simpler, giving the second form of the formula at the start of this section.

We can also write this in terms of 5 since Phi = 1 + 5 and –phi = 1 – 5 :
2
2

Fib(n) = Phin – (–phi)n = Phin–(–phi)n = 11 + √5n–1 – √5nPhi – (–phi)√5√522

If you prefer values in your formulae, then here is another form:-

Fib(n) = 1.6180339..n – (–0.6180339..)n
2.236067977..

This is a surprising formula since it involves square roots and powers of Phi (an irrational number) but it always gives an integer for all (integer) values of n!